The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self-Consistent Source

Abstract

The Painlevé property and Bäcklund transformation for the KdV equation with a self-consistent source are presented. By testing the equation, it is shown that the equation has the Painlevé property. In order to further prove its integrality, we give its bilinear form and construct its bilinear Bäcklund transformation by the Hirota's bilinear operator. And then the soliton solution of the equation is obtained, based on the proposed bilinear form.

1. Introduction

It is well known that some nonlinear partial differential equations such as the soliton equations with self-consistent sources have important physical applications. In recent years, there are many ways for solving the soliton equations that can be used to the soliton equations with self-consistent sources as well. For example, the soliton solutions of some equations such as the KdV, AKNS, and nonlinear schrödinger equation with self-consistent sources are obtained through the inverse scattering method [1, 2]. In [3] a Darboux transformation, positon and negaton solutions to a Schrödinger self-consistent source equation are further constructed. Also, the binary Darboux transformations for the KdV hierarchies with self-consistent sources were presented in [4]. In addition to that, the Hirota bilinear method has been successfully used in the search for exact solutions of continuous and discrete systems, and also in the search for new integrable equations by testing for multisoliton solutions or Bäcklund transformation [5, 6]. Recently a bilinear Bäcklund transformation has been presented for a (3+1)-dimensional generalized KP equation. Meanwhile, two classes of exponential and rational traveling wave solutions with arbitrary wave numbers are computed by applying the proposed bilinear Bäcklund transformation (see [7] for details). It is a good reference for solving many high-dimensional soliton equations.

Besides, the Painlevé analysis is a powerful tool for identifying the integrability of a nonlinear system. A partial different equation has the Painlevé property when the solutions of the partial different equation are single-valued about the movable, singularity manifold [8]. The basic thought as follows: if the singularity manifold is determined by 𝜙(𝑧₁,𝑧₂,…,𝑧_𝑛)=0 and 𝑢=𝑢(𝑧₁,𝑧₂,…,𝑧_𝑛) is a solution of the partial different equation, then we assume that𝑢=𝜙^𝛼∞_𝑗=0𝑢_𝑗𝜙^𝑗, (1.1) where𝜙=𝜙𝑧₁,𝑧₂,…,𝑧_𝑛,𝑢_𝑗=𝑢_𝑗𝑧₁,𝑧₂,…,𝑧_𝑛,𝑢₀≠0, (1.2)

are analytic functions of (𝑧_𝑗) in a neighborhood of the manifold 𝜙=0 and 𝛼 is an integer. Substitution of (1.1) into the partial different equation determines the values of 𝛼 and defines the recursion relations for 𝑢_𝑗,𝑗≠0,1,2,⋯. When the anatz (1.1) is correct, the pde is said to possess the Painlevé property and is conjectured to be integrable [9].

Motivated by the previous works, we focus our attention on the following nonlinear partial differential equations (PDEs) which is expressed by𝑢_𝑡+𝑢_𝑥𝑥𝑥+12𝑢𝑢_𝑥=𝜑²_𝑥,𝜑_𝑥𝑥+2𝑢𝜑=𝜆𝜑, (1.3) where 𝜆 is an arbitrary constant. In fact, (1.3) is a reduced form of the KdV equation with a source by symmetry constraints [10, 11]. The main purpose of this paper is to demonstrate the connection between the Painlevé property and the Bäcklund transformation for (1.3). Moreover, we get the bilinear Bäcklund transformation and the exact solution for (1.3) by the Hirota bilinear method. Thus we further convince the integrability of the equation.

The paper is organized as follows. In Section 2, we investigate the Painlevé property for (1.3). By testing the equation it is shown that the equation has the Painlevé property. Furthermore, we obtain a Bäcklund transformation of (1.3). In Section 3, using the Hirota’s bilinear operator, we obtain its bilinear form and construct its bilinear Bäcklund transformation. And then its one-soliton solution is obtained. Finally, conclusion is given in Section 4.

2. PainlEvÉ Test

As we know, the basic Painlevé test for ODEs consists of the following steps [12].

Step 1. Identify all possible dominant balances, that is, all singularities of form 𝑢∼𝑢₀(𝑧−𝑧₀)^𝜇.

Step 2. If all exponents 𝜇 are integers, find the resonances where arbitrary constants can appear.

Step 3. If all resonances are integers, check the resonance conditions in each Laurent expansion.

Conclusion. If no obstruction is found in Steps 1–3 for every dominant balances, then the Painlevé test is satisfied.
The above series may be substituted into the PDEs. Now we apply the above steps to (1.3). We will further give all possible solutions with integer resonances but without further analysis of the last cases. The expansions about the singular manifold have the forms:𝑢=^∞_𝑗=0𝑎_𝑗𝜙^𝑗+𝛼,𝜑=^∞_𝑗=0𝑏_𝑗𝜙^𝑗+𝛽. (2.1) To find the dominant balances, we are looking for leading order singular behaviour of the form 𝑢≈𝑎₀𝜙^𝛼,𝜑≈𝑏₀𝜙^𝛽. (2.2) And the derivatives of (2.2) are given by 𝑢_t≈𝑎₀𝛼𝜙^𝛼−1𝜙_𝑡,𝑢_𝑥≈𝑎₀𝛼𝜙^𝛼−1𝜙_𝑥,𝑢_𝑥𝑥𝑥≈𝑎₀𝛼(𝛼−1)(𝛼−2)𝜙^𝛼−3𝜙³_𝑥,𝜑_𝑥≈𝑏₀𝛽𝜙^𝛽−1𝜙_𝑥,𝜑_𝑥𝑥≈𝑏₀𝛽𝛽−1𝜙^𝛽−2𝜙²_𝑥. (2.3) Substituting (2.2)-(2.3) into (1.3), we get the following forms 𝑎₀𝛼(𝛼−1)(𝛼−2)𝜙^𝛼−3𝜙³_𝑥+12𝑎²₀𝛼𝜙_𝑥𝜙^2𝛼−1=2𝑏²₀𝛽𝜙_𝑥𝜙^2𝛽−1,𝑏₀𝛽𝛽−1𝜙^𝛽−2𝜙²_𝑥+2𝑎₀𝑏₀𝜙^𝛼+𝛽=0. (2.4) Calculating and simplifying equation (2.4), we get the dominant balances 𝑢≈−3𝜙²_𝑥𝜙⁻²,𝜑≈±6𝜙²_𝑥𝜙⁻², (2.5) where 𝛼=𝛽=−2,𝑎₀=−3𝜙²_𝑥,𝑏₀=±6𝜙²_𝑥. So we complete the first step.
The second step in applying the Painlevé test is to find the resonances. To find the resonances numbers 𝑗, we substitute (2.1) into (1.3), and collecting terms of each order of 𝜙, we obtain
𝜙⁻⁵:−24𝑎₀𝜙³_𝑥−24𝑎²₀𝜙_𝑥=−4𝑏²₀𝜙_𝑥. (2.6) By calculating equation (2.6), we get 𝑏²₀=6𝑎₀(𝑎₀+𝜙²_𝑥).
𝜙⁻⁴:18𝑎_0𝑥𝜙²_𝑥−6𝑎₁𝜙³_𝑥+18𝑎₀𝜙_𝑥𝜙_𝑥𝑥+12𝑎₀𝑎_0𝑥−36𝑎₀𝑎₁𝜙_𝑥=2𝑏₀𝑏_0𝑥−6𝑏₀𝑏₁𝜙_𝑥,6𝑏₀𝜙²_𝑥+2𝑎₀𝑏₀=0. (2.7) From (2.6) and (2.7), we have 𝑎₀=−3𝜙²_𝑥, 𝑏₀=±6𝜙²_𝑥, 6b₁=±(15𝜙_𝑥𝑥−17𝑎₁), where 𝑎₀ and 𝑏₀ are consistent with step one.
𝜙⁻³:−𝑎₀𝜙_𝑡−3𝑎_0𝑥𝑥𝜙_𝑥+3𝑎_1𝑥𝜙²_𝑥−3𝑎_0𝑥𝜙_𝑥𝑥+3𝑎₁𝜙_𝑥𝜙_𝑥𝑥−𝑎₀𝜙_𝑥𝑥𝑥+6𝑎₀𝑎_1𝑥+6𝑎₁𝑎_0𝑥−6𝑎²₁𝜙_𝑥−12𝑎₀𝑎₂𝜙_𝑥=𝑏₀𝑏_1𝑥+𝑏₁𝑏_0𝑥−𝑏²₁𝜙_𝑥−2𝑏₀𝑏₂𝜙_𝑥,−2𝑏_0𝑥𝜙_𝑥+𝑏₁𝜙²_𝑥−𝑏₀𝜙_𝑥𝑥+𝑎₀𝑏₁+𝑎₁𝑏₀=0. (2.8) From (2.7) and (2.8), we get 𝑎₁=3𝜙_𝑥𝑥, 𝑏₁=∓6𝜙_𝑥𝑥,
𝜙⁻²:𝑎_0𝑡−𝑎₁𝜙_𝑡+𝑎_{0𝑥𝑥𝑥}−3𝑎_1𝑥𝑥𝜙_𝑥−3𝑎_1𝑥𝜙_𝑥𝑥−𝑎₁𝜙_𝑥𝑥𝑥+12𝑎₀𝑎_2𝑥+12𝑎₁𝑎_1𝑥+12𝑎₂𝑎_0𝑥−12𝑎₀𝑎₃𝜙_𝑥−12𝑎₁𝑎₂𝜙_𝑥=2𝑏₀𝑏_2𝑥+2𝑏₁𝑏_1𝑥+2𝑏₂𝑏_0𝑥−2𝑏₀𝑏₃𝜙_𝑥−2𝑏₁𝑏₂𝜙_𝑥,𝑏_0𝑥𝑥−2𝑏_1𝑥𝜙_𝑥−𝑏₁𝜙_𝑥𝑥+2𝑎₀𝑏₂+2𝑎₁𝑏₁+2𝑎₂𝑏₀=𝜆𝑏₀. (2.9) From (2.8) and (2.9), we get 𝑎₂=15𝜙²_𝑥𝑥−20𝜙_𝑥𝜙_𝑥𝑥𝑥−𝜙_𝑥𝜙_𝑡+4𝜆𝜙²_𝑥20𝜙²_𝑥,𝑏₂=±−15𝜙²_𝑥𝑥+20𝜙_𝑥𝜙_𝑥𝑥𝑥−𝜙_𝑥𝜙_𝑡−6𝜆𝜙²_𝑥10𝜙²_𝑥, (2.10)
𝜙^𝑗−5:𝑎_{𝑗−3,𝑡}+𝑗−4𝑎_𝑗−2𝜙_𝑡+𝑎_{𝑗−3,𝑥𝑥𝑥}+3𝑗−4𝑎_{𝑗−2,𝑥𝑥}𝜙_𝑥+3𝑗−3𝑗−4𝑎_{𝑗−1,𝑥}𝜙²_𝑥+𝑗−2𝑗−3𝑗−4𝑎_𝑗𝜙³_𝑥+3𝑗−4𝑎_{𝑗−2,𝑥}𝜙_𝑥𝑥+3𝑗−3𝑗−4𝑎_𝑗−1𝜙_𝑥𝜙_𝑥𝑥+𝑗−4𝑎_𝑗−2𝜙_𝑥𝑥𝑥+12𝑎₀𝑎_{𝑗−1,𝑥}+𝑎₁𝑎_{𝑗−2,𝑥}+⋅⋅⋅+𝑎_𝑗−1𝑎_0,𝑥+𝜙_𝑥𝑗−2𝑎₀𝑎_𝑗+𝑗−3𝑎₁𝑎_𝑗−1+⋅⋅⋅+2𝑎_𝑗−4𝑎₄+𝑎_𝑗−3𝑎₃−𝑎_𝑗−1𝑎₁−2𝑎_𝑗𝑎₀=2𝑏₀𝑏_{𝑗−1,𝑥}+𝑏₁𝑏_{𝑗−2,𝑥}+⋅⋅⋅+𝑏_𝑗−1𝑏_0,𝑥+𝜙_𝑥𝑗−2𝑏₀𝑏_𝑗+𝑗−3𝑏₁𝑏_𝑗−1+⋯+2𝑏_𝑗−4𝑏₄+𝑏_𝑗−3𝑏₃−𝑏_𝑗−1𝑏₁−2𝑏_𝑗𝑏₀, (2.11)
𝜙^𝑗−4:𝑏_{𝑗−2,𝑥𝑥}+2𝑗−3𝑏_{𝑗−1,𝑥}𝜙_𝑥+𝑗−2𝑗−3𝑏_𝑗𝜙²_𝑥+j−3𝑏_𝑗−1𝜙_𝑥𝑥+2𝑎₀𝑏_𝑗+2𝑎₁𝑏_𝑗−1+⋯+𝑎_𝑗𝑏₀=𝜆𝑏_𝑗−2. (2.12) Substituting 𝑎₀=−3𝜙²_𝑥,𝑏₀=±6𝜙²_𝑥 into (2.11) and (2.12), the coefficients of 𝑎_𝑗 and 𝑏_𝑗 may be rearranged to give 𝑗−2𝑗−3−36𝑗−4𝑎_𝑗𝜙³_𝑥+𝐹𝑎₀,𝑎₁,…,𝑎_𝑗−1,𝜙_𝑡,𝜙_𝑥,𝜙_𝑥𝑥𝑏,…=±12𝑗−4_𝑗𝜙³_𝑥+𝐺𝑏₀,𝑏₁,…,𝑏_𝑗−1,𝜙_𝑡,𝜙_𝑥,𝜙_𝑥𝑥,𝑏,…𝑗−2𝑗−3−6_𝑗𝜙²_𝑥±12𝑎_𝑗𝜙²_𝑥+𝐻𝑎₀,𝑎₁,…,𝑎_𝑗−1,𝑏₀,𝑏₁,⋯,𝑏_𝑗−1,𝜙_𝑡,𝜙_𝑥,𝜙_𝑥𝑥,…=0, (2.13)(2.14) simplifying (2.13)-(2.14), we have 𝑗−4𝑗𝑗−5𝑗²−5𝑗−30+144𝑗²−5𝑗−30𝑎_𝑗𝜙³_𝑥=−𝑗𝑗−5𝑗²𝐹𝑗−5𝑗−30+𝑗𝑗−5²−5𝑗−30𝐺∓12𝑗−4𝑗²−5𝑗−30𝜙_𝑥𝑗𝑗𝐻,𝑗−4𝑗−5²−5𝑗−30+144𝑏_𝑗𝜙³_𝑥=−𝑗²−5𝑗−30𝑗−4𝜙_𝑥𝐻∓12𝐺±12𝐹. (2.15)(2.16) There, it is found that the resonance occurs at 𝑗=4, so the second step is completed.
For the last step, we will check the resonance conditions. So we need to find the orders in the expansion (2.1) where arbitrary constants may appear:
𝜙⁻¹:𝑎_1𝑡+𝑎_{1𝑥𝑥𝑥}+12𝑎₀𝑎_3𝑥+𝑎₁𝑎_2𝑥+𝑎₂𝑎_1𝑥+𝑎₃𝑎_0𝑥+𝜙_𝑥2𝑎₀𝑎₄+𝑎₁𝑎₃−𝑎₃𝑎₁−2𝑎₄𝑎₀=2𝑏₀𝑏_3𝑥+𝑏₁𝑏_2𝑥+𝑏₂𝑏_1𝑥+𝑏₃𝑏_0𝑥+𝜙_𝑥2𝑏₀𝑏₄+𝑏₁𝑏₃−𝑏₃𝑏₁−2𝑏₄𝑏₀,𝑏_1𝑥𝑥+2𝑎₀𝑏₃+2𝑎₁𝑏₂+2𝑎₂𝑏₁+2𝑎₃𝑏₀=𝜆𝑏₁. (2.17) From (2.17), we know 𝑎₄ and 𝑏₄ are both arbitrary. Thus (1.3) possess the Painlevé property.
We now specialize (2.1) by setting the resonance functions 𝑎₄=𝑏₄=0. Furthermore, we require 𝑎₃=𝑏₃=0, it is easily demonstrated that 𝑎_𝑗=0,𝑏_𝑗=0,𝑗≥3 from the recursion relations.
If 𝑎₂ and 𝑏₂ satisfy𝑎_2𝑡+𝑎_{2𝑥𝑥𝑥}+12𝑎₂𝑎_2𝑥=𝑏²_2𝑥,𝑏_2𝑥𝑥+2𝑎₂𝑏₂=𝜆𝑏₂, (2.18) we obtain a Bäcklund transformation of (1.3): 𝑢=3ln𝜙_𝑥𝑥+𝑢,𝑢≡𝑎₂,𝜑=−6ln𝜙_𝑥𝑥+𝜑,𝜑≡𝑏₂, (2.19) where we consider the case of 𝑏₀=6𝜙²_𝑥, 𝑏₁=−6𝜙_𝑥𝑥 moreover, 𝑢,𝜑 and 𝑢,𝜑 satisfy (1.3) and 3𝑢ln𝜙_𝑥𝑥+𝜑ln𝜙_𝑥𝑥(𝑡),𝜙=𝜃_𝑡𝜙_𝑥=5𝜆+15𝜙²_𝑥𝑥𝜙²_𝑥−20𝜙_𝑥𝑥𝑥𝜙_𝑥. (2.20) Many studies [9, 13] show that a new solution can usually be obtained from a given solution of an equation if the so-called Bäcklund transformation for the equation is found. Therefore, it is worth to find the Bäcklund transformation of an equation. In the next section, we will give the bilinear Bäcklund transformation of (1.3).

3. Bilinear Form

As we know, when you want to use Hirota method, the first thing you need to do is to rewrite the equation under consideration as the bilinear form [14]. This can be achieved for (1.3) by the following dependent variable transformation:𝑢=ln𝑓_𝑥𝑥𝑔,𝜑=𝑓. (3.1) Equation (1.3) can be written into bilinear forms𝐷_𝑥𝐷_𝑡+𝐷⁴_𝑥𝑓⋅𝑓=2𝑔²,𝐷²_𝑥𝑓⋅𝑔=𝜆𝑓𝑔, (3.2) where 𝐷 is the well-known Hirota bilinear operator𝐷^𝑚_𝑥𝐷^𝑛_𝑡𝜕𝑎⋅𝑏=𝜕𝑥₁−𝜕𝜕𝑥₂^𝑚𝜕𝜕𝑡₁−𝜕𝜕𝑡₂^𝑛𝑎𝑥₁,𝑡₁𝑏𝑥₂,𝑡_{2𝑥1=𝑥2,𝑡1=𝑡2.}. (3.3)

Now we will give the bilinear Bäcklund transformation of (1.3).

Theorem 3.1. Suppose that (𝑓, 𝑔) is a solution of (3.2), then (𝑓^′, 𝑔^′), satisfying the following relations: 𝐷_𝑥𝐷𝑔⋅𝑓′−𝜇𝑓𝑔′=0,_𝑥𝐷𝑓⋅𝑔′−𝜇𝑔𝑓′=0,_2𝑥𝐷−𝜈𝑓⋅𝑓′=0,_𝑡+𝐷³_𝑥+3𝜈𝐷_𝑥1𝑓⋅𝑓′+𝜇𝑔𝑔′=0, (3.4)(3.5)(3.6)(3.7) is another solution of (3.2), where 𝜇 and 𝜈 are arbitrary constants and 𝜇≠0.

Proof. We consider the following: 𝑃₁=𝐷_𝑥𝐷_𝑡+𝐷⁴_𝑥𝑓^′⋅𝑓^′−2𝑔^′2,𝑃₂=𝐷²_𝑥𝑓^′⋅𝑔^′−𝜆𝑓^′𝑔^′. (3.8) We will show that (3.4)–(3.7) imply 𝑃₁=0 and 𝑃₂=0. We first work on the case of 𝑃₁. We will use various bilinear identities which, for convenience, are presented in the appendix: −𝑃₁𝑓𝑓=𝐷_𝑥𝐷_𝑡+𝐷⁴_𝑥𝑓⋅𝑓−2𝑔²𝑓′𝑓′−𝐷_𝑥𝐷_𝑡+𝐷⁴_𝑥𝑓^′⋅𝑓^′−2𝑔^′2𝑓𝑓_(A.1)(A).2=2𝐷_𝑥𝐷_𝑡𝑓⋅𝑓^′⋅𝑓𝑓′+2𝐷_𝑥𝐷³_𝑥𝑓⋅𝑓^′⋅𝑓𝑓^′+3𝐷²_𝑥𝑓⋅𝑓^′⋅𝐷_𝑥𝑓^′⋅𝑓+2𝑔′𝑔′𝑓𝑓−2𝑔𝑔𝑓′𝑓′_()3.6=2𝐷_𝑥𝐷_𝑡𝑓⋅𝑓^′⋅𝑓𝑓′+2𝐷_𝑥𝐷³_𝑥𝑓⋅𝑓^′⋅𝑓𝑓^′+3𝜈𝑓𝑓^′⋅𝐷_𝑥𝑓^′⋅𝑓+2𝑓𝑔′𝑓𝑔′−2𝑔𝑓′𝑔𝑓′_(3.4)(3.5)=2𝐷_𝑥𝐷_𝑡+𝐷³_𝑥+3𝜈𝐷_𝑥𝑓⋅𝑓^′1⋅𝑓𝑓′+2𝜇𝐷_𝑥𝑔⋅𝑓^′𝑓𝑔^′−𝐷_𝑥𝑓⋅𝑔^′𝑔𝑓^′_(A.3)=2𝐷_𝑥𝐷_𝑡+𝐷³_𝑥+3𝜈𝐷_𝑥𝑓⋅𝑓^′1⋅𝑓𝑓′+2𝜇𝐷_𝑥𝑔𝑔′⋅𝑓𝑓′=2𝐷_𝑥𝐷_𝑡+𝐷³_𝑥+3𝜈𝐷_𝑥𝑓⋅𝑓^′+1𝜇𝑔𝑔^′⋅𝑓𝑓′_(3.7)≡0. (3.9)Next we come to the second part of the proof: −𝑃₂𝑓𝑔=𝐷²_𝑥𝑓⋅𝑔−𝜆𝑓𝑔𝑓′𝑔′−𝐷²_𝑥𝑓^′⋅𝑔^′−𝜆𝑓^′𝑔^′𝑓𝑔_(A.4)=𝐷_𝑥𝐷_𝑥𝑓⋅𝑔^′⋅𝑓^′𝑔+𝑓𝑔^′⋅𝐷_𝑥𝑓^′⋅𝑔_()3.4)(3.5=𝐷_𝑥𝜇𝑔𝑓^′⋅𝑓^′𝑔−𝑓𝑔^′⋅𝜇𝑓𝑔≡^′(A.5)0. (3.10) Thus we have completed the proof of Theorem 3.1.

We will show that our Bäcklund transformation (3.4)–(3.7) supplies us with a Lax representation for (1.3). Suppose𝑣=ln𝑓_𝑥𝑥,𝑔=𝜑𝑓,𝑓^′=𝑋𝑓,𝑔^′=𝑌𝑔, (3.11) then the variables 𝑓, 𝑔, 𝑓′ and 𝑔′ can be eliminated from (3.4)–(3.7). The elimination results in𝑋_𝑥=𝜑_𝑥𝜑𝑌𝑋−𝜇𝑌,_𝑥𝜑=−^𝑥𝜑𝑋𝑌−𝜇𝑋,_𝑥𝑥=()𝑋𝜈−2𝜔𝑋,_𝑡=−𝑋_𝑥𝑥𝑥−3(2𝜔+𝜈)𝑋_𝑥+1𝜇𝜑²𝑌. (3.12)(3.13)(3.14)(3.15) Therefore, we have the following.

Theorem 3.2. The compatibility condition of (3.12)–(3.15) is (1.3). In fact, using the compatibility conditions 𝑋_𝑥𝑥𝑡=𝑋_𝑡𝑥𝑥, one can obtain (1.3) where 𝜈 and 𝜔 satisfy 𝜈−2𝜔=𝜆.

Finally we will give the soliton solution of the equation (1.3) by the standard perturbation method:𝑢=ln1+𝑒^2𝜉_𝑥𝑥,𝜑=2𝑘𝛽(𝑡)𝑒^𝜉1+𝑒^2𝜉, (3.16)

where 𝜉=𝑘𝑥−4𝑘³𝑡+∫^𝑡₀𝛽(𝑠)𝑑𝑠+𝑐, 𝑐, 𝑘 are arbirtary constants and 𝜆=𝑘².

4. Conclusion

In this paper, we investigate the Painlevé property for the KdV equation with a self-consistent source. By tests to the equation, it is shown that only the principal balance of the equation has the Painlevé property. While noninteger resonances are allowed with the weak extension of the Painlevé test [12]. We obtain the two different Bäcklund transformations. And then the soliton solution for (1.3) is given.

Appendix

In this appendix, we list the relevant bilinear identities, which can be proved directly. Here 𝑎, 𝑏, 𝑐, and 𝑑 are arbitrary functions of the independent variables 𝑥 and 𝑡∶𝐷_𝑥𝐷_𝑡𝑎⋅𝑎𝑏²−𝑎²𝐷_𝑥𝐷_𝑡𝑏⋅𝑏=2𝐷_𝑥𝐷_𝑡𝑎⋅𝑏⋅𝑏𝑎, (A.1)𝐷⁴_𝑥𝑎⋅𝑎𝑏²−𝑎²𝐷⁴_𝑥𝑏⋅𝑏=2𝐷_𝑥𝐷³_𝑥𝑎⋅𝑏⋅𝑎𝑏+3𝐷²_𝑥𝑎⋅𝑏⋅𝐷_𝑥𝑏⋅𝑎, (A.2)𝐷_𝑥𝑎⋅𝑏𝑐𝑑−𝑎𝑏𝐷_𝑥𝑐⋅𝑑=𝐷_𝑥(𝑎𝑑)⋅(𝑐𝑏), (A.3)𝐷²_𝑥𝑎⋅𝑏𝑐𝑑−𝑎𝑏𝐷²_𝑥𝑐⋅𝑑=𝐷_𝑥𝐷_𝑥𝑎⋅𝑑⋅𝑐𝑏+𝑎𝑑⋅𝐷_𝑥𝑐⋅𝑏, (A.4)𝐷_𝑥𝑎⋅𝑎=0. (A.5)

Acknowledgments

This work was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young Foundation of Shanxi Province (no. 2011021001-1), Research Project Supported by Shanxi Scholarship Council of China (2011-093), and the Major Subject Foundation of Shanxi.